For information to anyone with the same problem: eventually I ordered a new printhead from Ebay, I think it was shipped from PRC to me in Finland, so it took some time to get it, but the cost was not huge: it cost a bit less than 40 USD, including shipping and sales tax. When I got the printhead, it was easy for me to replace the old printhead with the new one. I did this already some time ago, so I don't remember exactly how it went, but iirc, when I installed the new printhead, the printer did something for couple of minutes, before it was ready to be used. And, after that, luckily my problem was solved, so the new printhead indeed seems to have fixed my problem! So, now the big black ink cartridge is working again.
So, like I already wrote, if you have similar problem as I had, it's possible to fix by yourself, when you just buy a new printhead.
Oh, and btw. The only reply I got to my message here was from moderator stating that this forum is Canon USA and they can't help me, because I'm not residing in USA, and I should contact my local Canon support. Well, I really did contact Canon Finland support. Unfortunately they don't have a nice forum like this, so I was just able to e-mail them, and their reply was pretty useless to me: they just told me that I have obsolete printer, to which they don't have any spare parts anymore, so I should just buy a new Canon printer. They gave me (or offered at least) a 20 % discount coupon to buy a new Canon printer. I'm happy I didn't value that answer too much, but instead decided to try if I was able to fix the printer myself. This fix was so easy, simple and cheap, that I think that it would have been absolute waste of resources to buy a new printer, and consequently of course throw the old printer into trash.
So, I really didn't come here to thank anyone, as I was able to solve the problem myself, but to give information to anyone with the same problem, that it's possible to fix it (regardless of what Canon might be saying).
Thank You.
